# Logical disjunction

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In logic and mathematics, or, also known as logical disjunction or inclusive disjunction, is a logical operator that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are true. In grammar, or is a coordinating conjunction. In ordinary language "or" sometimes has the meaning of exclusive disjunction.

## Contents

### Notation

Or is usually expressed with an infix operator. In mathematics and logic, it is usually ; in electronics, +; and in programming languages, | or or. Some programming languages have a related control structure, the short-circuit or, written ||, or else, etc.

### Definition

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false. More generally a disjunction is a logical formula that can have one or more literals separated only by ORs. A single literal is often considered to be a degenerate disjunction.

### Truth table

The truth table of $~A \or B$:

### Properties

The following properties apply to disjunction:

• associativity: $a \lor (b \lor c) \equiv (a \lor b) \lor c$
• commutativity: $a \lor b \equiv b \lor a$